The research in this thesis is aimed at addressing this issue by studying the properties and vulnerabilities of existing methods, developing new methods to overcome existing limitations, and applying new methods to biomedically relevant applications. By studying the connection among existing methods, an integrative framework was developed that facilitated the design of new reconstruction methods in a systematic fashion. One method in particular, Broad-use Linear Acquisition Speed-up Technique (BLAST), showed considerable improvement over existing methods in terms of reconstruction accuracy and its broad applicability. In addition to introducing several new reconstruction methods, this research led to the development of a consensus reconstruction approach, which adaptively combined several reconstruction methods together to improve the overall robustness. The feasibility of these reconstruction methods and approaches was demonstrated with a wide range of applications, including reduced field-of-view dynamic imaging, T2 mapping, MR thermometry, and MR spectroscopic imaging.